Whether $||a||_{L^p} +||b||_{L^p}\le||\sqrt{a^2+b^2}||_{L^p}$

Question

$a=a(x), b=b(x)$ are elements of $L^p(\Omega)$, $\Omega$ is bounded open subset of $R^n$.

Whether $||a||_{L^p} +||b||_{L^p}\le||\sqrt{a^2+b^2}||_{L^p}$ ?

Answer 1

No way. Let $p=1$, and take constant functions $a=3$, and $b=4$. $3+4 > 5.$